279 
an expansion of the form Fim in a number of terms each of which 
is a sum of prodnets of a non-reduecible form with a certain number 
(characteristic of that term) of determinants formed by n ofthe sets 
of variables. 
For mn this expansion has been given first by Caprini') and 
for the general case by J. Deruyts*) and K. Perr®)*). Both Capen 
and Prrr base their proof upon the property mentioned p. 276, 
that a form in 7 sets of n variables containing the determinant of 
the variables as a factor, is not annihilated by (2. The deduction 
of CaprELii, which is most analogous to the above is based upon the 
theory of the differential operators Hs). Drruyrs uses his theory of 
the semi-invariants and -covariants and Perr makes use of differential- 
operators that can be constructed by means of auxiliary variables. 
The terms of the expansion in non-reducible covariants may 
P 
again be decomposed in different ways. Firstly each term of m 
can be decomposed into general mixed alternations and these again 
P 
into simple ones. To this corresponds an expansion of m in a sum 
of terms consisting each of a sum of products of a number of 
determinants with s,,..., s,; rows formed from the characteristic numbers 
of the sets x’, y’... with one single symmetrical form. All terms are 
covariants, the sub-terms only then when s,;—=s,=...=n, 
In each sub-term the power of a determinant of the characteristic 
numbers of the variables is the same as the power of the determinant 
of the characteristic numbers of the corresponding ideal factors of 
yc 
_m. Under these conditions the expansion is singly determined and 
an extension of the one given by Rerssincrr for n — 2. 
P 
Secondly each term of m may be decomposed into ordered alternations 
of the form ,,A. Then the determinants occurring in each sub-term 
must belong to the permutation regions of a definite ordered alter- 
nation „„4A, characteristic of that sub-term, and acting on x’e vin 
1) Fondamenti di una teoria generale delle forme algebriche, Mem. dei Linces 
(82) § 74; Sur les opérations dans la théorie des formes algébriques, Math Ann. 
37 (90) 1—37. 
*) Essai d'une théorie générale des formes algébriques. Mém de Liège. 2. 17 
(92) 4. 1—156; Détermination des fonctions invariantes de formes a plusieurs 
séries de variables. Mém. couronnés et mém. des sav. étr. de Bruxelles 53 (90— 
93) 2. 1—23. 
5) Ueber eine Reihenentwicklung für algebraische Formen, Bull. Intern. de Prague 
12 (07) 168—191. 
*) The forms called here non-reduceable are called by Capetu: “formes impro- 
pres” and by Deruyts: “covariants de formes primaires”. 
